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In [[mathematics]], the '''uniform boundedness principle''' or '''Banach–Steinhaus theorem''' is one of the fundamental results in [[functional analysis]]. Together with the [[Hahn–Banach theorem]] and the [[open mapping theorem (functional analysis)|open mapping theorem]], it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of [[continuous linear operator]]s (and thus bounded operators) whose domain is a [[Banach space]], pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by [[Stefan Banach]] and [[Hugo Steinhaus]] but it was also proven independently by [[Hans Hahn (mathematician)|Hans Hahn]].

== Uniform boundedness principle ==
<blockquote>'''Theorem (Uniform Boundedness Principle).''' Let ''X'' be a [[Banach space]] and ''Y'' be a [[normed vector space]]. Suppose that ''F'' is a collection of continuous linear operators from ''X'' to ''Y''. If for all ''x'' in ''X'' one has

:<math>\sup\nolimits_{T \in F} \|T(x)\|_Y < \infty, </math>

then

:<math>\sup\nolimits_{T \in F} \|T\|_{B(X,Y)} < \infty.</math></blockquote>

The completeness of ''X'' enables the following short proof, using the [[Baire category theorem]]. 

'''Proof.''' Suppose that for every ''x'' in the Banach space ''X'', one has:

:<math>\sup\nolimits_{T \in F} \|T (x)\|_Y < \infty.</math>

For every integer ''n'' in '''N''', let

:<math> X_n = \left \{x \in X \ : \ \sup\nolimits_{T \in F} \|T (x)\|_Y \le n \right \}. </math>

The set <math>X_n</math> is a [[closed set]] and by the assumption,

:<math>\bigcup\nolimits_{n \in \mathbf{N}} X_n = X \neq \varnothing.</math>

By the [[Baire category theorem]] for the non-empty [[complete metric space]] ''X'', there exists ''m'' such that
<math> X_m</math> has non-empty [[Interior (topology)|interior]], ''i.e.'', there exist <math>x_0 \in X_m</math> and {{nowrap|ε > 0}} such that

:<math> \overline{B_\varepsilon (x_0)} := \{x \in X \,:\, \|x - x_0\| \le \varepsilon \} \subseteq X_m.</math>

Let ''u'' ∈ ''X'' with {{nowrap|ǁ''u''ǁ ≤ 1}} and {{nowrap|''T'' ∈ ''F''}}. One has that:

:<math>\begin{align}
\|T(u) \|_Y &= \varepsilon^{-1} \left \|T \left( x_0 + \varepsilon u \right) - T(x_0) \right \|_Y & [\text{by linearity of } T ] \\
&\leq \varepsilon^{-1} \left ( \left\| T (x_0 + \varepsilon u) \right\|_Y + \left\| T (x_0) \right\|_Y \right ) \\
&\leq \varepsilon^{-1} (m + m). & [ \text{since} \ x_0 + \varepsilon u, \ x_0 \in X_m ] \\
\end{align}</math>

Taking the supremum over ''u'' in the unit ball of ''X'', it follows that

:<math> \sup\nolimits_{T \in F} \|T\|_{B(X,Y)} \leq 2 \varepsilon^{-1} m < \infty.</math>

==Corollaries==
<blockquote>'''Corollary.''' If a sequence of bounded operators (''Tn'') converges pointwise, that is, the limit of {''Tn''(''x'')} exists for all ''x'' in ''X'', then these pointwise limits define a bounded operator ''T''.</blockquote>

Note it is not claimed above that ''Tn'' converges to ''T'' in operator norm, i.e. uniformly on bounded sets. (However, since {''Tn''} is bounded in operator norm, and the limit operator ''T'' is continuous, a standard [[3-epsilon estimate|"3-ε" estimate]] shows that ''Tn'' converges to ''T'' uniformly on ''compact'' sets.)

<blockquote>'''Corollary.''' Any weakly bounded subset S in a normed space Y is bounded''</blockquote>

Indeed, the elements of ''S'' define a pointwise bounded family of continuous linear forms on the Banach space ''X'' = ''Y*'', continuous dual of ''Y''. By the uniform boundedness principle, the norms of elements of ''S'', as functionals on ''X'', that is, norms in the second dual ''Y**'', are bounded. But for every ''s'' in ''S'', the norm in the second dual coincides with the norm in ''Y'', by a consequence of the [[Hahn–Banach theorem]].

Let ''L''(''X'', ''Y'') denote the continuous operators from ''X'' to ''Y'', with the operator norm. If the collection ''F'' is unbounded in ''L''(''X'', ''Y''), then by the uniform boundedness principle, we have:

:<math> R = \left \{ x \in X \ : \ \sup\nolimits_{T \in F} \|Tx\|_Y = \infty \right \} \neq \varnothing</math>

In fact, ''R'' is dense in ''X''. The complement of ''R'' in ''X'' is the countable union of closed sets ∪''Xn''. By the argument used in proving the theorem, each ''Xn'' is [[nowhere dense]], i.e. the subset ∪''Xn'' is ''of first category''. Therefore ''R'' is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called ''residual sets'') are dense. Such reasoning leads to the '''principle of condensation of singularities''', which can be formulated as follows:

<blockquote>'''Theorem.''' Let ''X'' be a Banach space, {''Yn''} a sequence of normed vector spaces, and ''Fn'' a unbounded family in ''L''(''X'', ''Yn''). Then the set

:<math> R = \left \{ x \in X \ : \ \forall n \in \mathbf{N} : \sup\nolimits_{T \in F_n} \|Tx\|_Y = \infty \right \}</math>

is dense in ''X''.</blockquote>

'''Proof.''' The complement of ''R'' is the countable union

:<math>\bigcup\nolimits_{n,m} \left \{ x \in X \ : \ \sup\nolimits_{T \in F_n} \|Tx\|_Y \le m \right \}</math>

of sets of first category. Therefore its residual set ''R'' is dense.

==Example: pointwise convergence of Fourier series==
Let '''T''' be the [[circle group|circle]], and let ''C''('''T''') be the Banach space of continuous functions on '''T''', with the [[uniform norm]]. Using the uniform boundedness principle, one can show that the Fourier series, "typically", does not converge pointwise for elements in ''C''('''T''').

For ''f'' in ''C''('''T'''), its [[Fourier series]] is defined by

:<math>\sum_{k \in \mathbf{Z}} \hat{f}(k) e^{ikx} = \sum_{k \in \mathbf{Z}}\frac{1}{2\pi} \left (\int_0 ^{2 \pi} f(t) e^{-ikt} dt \right) e^{ikx},</math>

and the ''N''-th symmetric partial sum is

:<math> S_N(f)(x) = \sum_{k=-N}^N \hat{f}(k) e^{ikx} = \frac{1}{2 \pi} \int_0 ^{2 \pi} f(t) D_N(x - t) \, dt,</math>

where ''DN'' is the ''N''-th [[Dirichlet kernel]]. Fix ''x'' in '''T''' and consider the convergence of {''SN''(''f'')(''x'')}. The functional φ''N,x'' : ''C''('''T''') → '''C''' defined by

:<math>\varphi_{N, x} (f) = S_N(f)(x), \qquad f \in C(\mathbf{T}),</math>

is bounded. The norm of φ''N,x'', in the dual of ''C''('''T'''), is the norm of the signed measure (2π)−1''D''''N''(''x''−''t'') d''t'', namely

:<math> \left \| \varphi_{N,x} \right \| = \frac{1}{2 \pi} \int_0 ^{2 \pi} \left | D_N(x-t) \right | \, dt = \frac{1}{2 \pi} \int_0 ^{2 \pi} \left | D_N(s) \right | \, ds = \left \| D_N \right \|_{L^1(\mathbf{T})}.</math>

One can verify that

:<math>\frac{1}{2 \pi} \int_0 ^{2 \pi} | D_N(t) | \, dt \ge \int_0^\pi \frac{\left |\sin\left ( (N+\tfrac{1}{2})t \right )\right|} t \, dt \to \infty.</math>

So the collection {φ''N,x''} is unbounded in ''C''('''T''')*, the dual of ''C''('''T'''). Therefore by the uniform boundedness principle, for any ''x'' in '''T''', the set of continuous functions whose Fourier series diverges at ''x'' is dense in ''C''('''T''').

More can be concluded by applying the principle of condensation of singularities. Let {''xm''} be a dense sequence in '''T'''. Define φ''N,xm'' in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each ''xm'' is dense in ''C''('''T''') (however, the Fourier series of a continuous function ''f'' converges to ''f''(''x'') for almost every ''x'' in '''T''', by [[Carleson's theorem]]).

== Generalizations ==
The least restrictive setting for the uniform boundedness principle is a [[barrelled space]] where the following generalized version of the theorem holds {{harv|Bourbaki|1987|loc=Theorem III.2.1}}:

<blockquote>'''Theorem.''' Given a barrelled space ''X'' and a [[locally convex space]] ''Y'', then any family of pointwise bounded [[continuous linear mapping]]s from ''X'' to ''Y'' is [[equicontinuous]] (even [[uniformly equicontinuous]]).</blockquote>

Alternatively, the statement also holds whenever ''X'' is a [[Baire space]] and ''Y'' is a locally convex space {{harv|Shtern|2001}}.

{{harvtxt|Dieudonné|1970}} proves a weaker form of this theorem with [[Fréchet space]]s rather than the usual Banach spaces. Specifically,

<blockquote>'''Theorem.''' Let ''X'' be a Fréchet space, ''Y'' a normed space, and ''H'' a set of continuous linear mappings of ''X'' into ''Y''. If for every ''x'' in ''X''
:<math>\sup\nolimits_{u\in H}\|u(x)\|<\infty,</math>
then the family ''H'' is equicontinuous.</blockquote>

==See also==
*[[Barrelled space]], a [[topological vector space]] with minimum requirements for the Banach Steinhaus theorem to hold

== References ==
*{{citation|first1=Stefan|last1=Banach|authorlink1=Stefan Banach|first2=Hugo|last2=Steinhaus|authorlink2=Hugo Steinhaus| url=http://matwbn.icm.edu.pl/ksiazki/fm/fm9/fm918.pdf |title=Sur le principe de la condensation de singularités|journal=[[Fundamenta Mathematicae]]| volume=9| pages=50–61|year=1927}}. {{fr}}
*{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|series=Elements of mathematics|title=Topological vector spaces|publisher=Springer|year=1987|isbn=978-3-540-42338-6}}
*{{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on analysis, Volume 2|year=1970|publisher=Academic Press}}.
*{{citation|first=Walter|last=Rudin|authorlink=Walter Rudin|title=Real and complex analysis|publisher=McGraw-Hill|year=1966}}.
*{{springer|Banach–Steinhaus theorem|first=A.I.|last=Shtern|year=2001|id=b/b015200}}.

[[Category:Functional analysis]]
[[Category:Articles containing proofs]]
[[Category:Mathematical principles]]
[[Category:Theorems in functional analysis]]

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